The critical points of the generalized complementary energy variational principles are clarified. An open problem left by Hellinger and Reissner is solved completely. A pure complementary energy (involving the Kirchhoff type stress only) is constructed. We prove that the well-known generalized Hellinger-Reissner’s energy L(u, s) is a saddle point functional if and only is the Gao-Strang gap fun...

Variational principle for probabilistic learning Yet another justification More simplification of updates for mean-field family Examples Dirichlet Process Mixture On minimization of divergence measures Energy minimization justifications Variational learning with exponential family Mean parametrization and marginal polytopes Convex dualities The log-partition function and conjugate duality Belie...

In this paper we study a non-homogeneous eigenvalue problem involving variable growth conditions and a sign-changing potential. We prove that any λ > 0 sufficiently small is an eigenvalue of the nonhomogeneous eigenvalue problem { −div(a(|∇u|)∇u) = λV(x)|u|q(x)−2u, in Ω, u = 0, on ∂Ω. The proofs of the main results are based on Ekeland’s variational principle.

The critical points of the generalized complementary energy variational principles are clariied. An open problem left by Hellinger and Reissner is solved completely. A pure complementary energy (involving the Kirchhoo type stress only) is constructed. We prove that the well-known generalized Hellinger-Reissner's energy L(u; s) is a saddle point functional if and only is the Gao-Strang gap funct...